Reporting back:
1. Catching a bus was pretty straightforward. I think bus #100 gets you from the Mokpo ferry Terminal to the Bus Terminal, or you could take a taxi for about 3-4000 Won. It was another 3,500 Won to get to Yeongam. The bus leaves for Yeongam every 20 minutes, and takes about an hour(ish) to get there.
2. Don't stay at the Regency Hotel. It's really close to the Yeongam Bus Terminal, but that's about it. It's 25,000 Won for a single person, 50,000 Won for a double with two people. And they don't take pets. It looks clean from the outside, I will admit.
We didn't stay there though.
A little bit farther down the street, past the Regency Hotel, there are a couple of yeogwans/motels. The one I can recommend is Motel Samhojang (삼호장). One of the proprietors there speaks English. The four of us (and a small dog) stayed there for only 40,000 Won total, which was much cheaper than the other places. We had a Korean-style ondol room. The place was nothing specail, but it was nice and clean and economical. The number for 삼호장 is (061)471-0067~9 if you're interested.
There was another motel close-by that we checked out, but it looked rather scuzzy. So make sure you head to 삼호장 and not that other place!
3. Alex, you're definitely right about being tired climbing up Wolchusan. It's not an especially tall mountain, but you just keep going up and down and up and down on your way to the peak. So you actually end up climbing much more than you'd think! But the little dog our friend brought made it to the peak and down with minimal carrying, so it wasn't too bad. Not sure if you're allowed to bring dogs into the peak, but nobody stopped us, and I think he was a hit with the other hikers.
I didn't get a chance to check out the hotsprings (although I did see signs).
We did head to the five-day market and picked up a rather tasty watermelon. The market's not all that huge, but nice nonetheless. The Yeongam market occurs on the 5th and 10th (and 15th, 20th, 25th, and so on).
4. I saw the path Alex mentioned (I think). There were two. One was labeled with a sign, and another one was a narrow unbeaten path wandering up into the hills. I really liked the latter path, but my friends wanted to go on one with some assurance that we'd make it to the peak, so we went the more popular route. There were a fair amount of people, but it wasn't too crowded.
All in all was had a great time.
Special note for anyone going to take a ferry to or from Jeju:
It can get pretty rough, especially with 80 km/h gusts. If you get seasick easily, just be warned.